There are different topological representations of non-orientable matroids. In this talk, inspired by Swartz's work, I will show an explicit homotopy sphere arrangement that is a regular CW complex whose intersection lattice is the geometric lattice of the corresponding matroid. We will also look at enumerative properties, including how the flag $f$-vector formula of Billera, Ehrenborg and Readdy for oriented matroids applies to arbitrary matroids.